On domination numbers of graph bundles

Abstract

Letγ(G) be the domination number of a graphG. It is shown that for anyκ ≥ 0 there exists a Cartesian graph bundleB█φF such thatγ(B█φF) =γ(B)γ(F) — 2κ. The domination numbers of Cartesian bundles of two cycles are determined exactly when the fibre graph is a triangle or a square. A statement similar to Vizing’s conjecture on strong graph bundles is shown not to be true by proving the inequalityγ(B █ φF) ≤γ(B)γ(F) for strong graph bundles. Examples of graphsB andF withγ(B █ φF)γ(B)γ(F) are given.